Actually, most public health policies have to do with work environments. If it involves a person's body at all, it typically involves their behavior in the work place.
As for kids, hell yes, I think parents should make the right choice. Do I feel like I should demand that of them? Not really. To an extent, it may fall under neglect and I would back such a policy if it came about, but only for certain vaccinations. Low risk with a high probability of infection without a vaccination may constitute neglect in my mind if the parent withholds the vaccination. However, something like the flu vaccine is not high risk for a healthy child. Most children can fight off the flu with no consequences. And . . . yes, the worst aspects of the flu can be treated with antibiotics.
As for kids, hell yes, I think parents should make the right choice. Do I feel like I should demand that of them? Not really. To an extent, it may fall under neglect and I would back such a policy if it came about, but only for certain vaccinations. Low risk with a high probability of infection without a vaccination may constitute neglect in my mind if the parent withholds the vaccination. However, something like the flu vaccine is not high risk for a healthy child. Most children can fight off the flu with no consequences. And . . . yes, the worst aspects of the flu can be treated with antibiotics.